### Abstract

Original language | English |
---|---|

Pages (from-to) | 413 - 434 |

Number of pages | 22 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 34 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*34*(4), 413 - 434. https://doi.org/10.1007/s00526-008-0189-y

}

*Calculus of Variations and Partial Differential Equations*, vol. 34, no. 4, pp. 413 - 434. https://doi.org/10.1007/s00526-008-0189-y

**Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions.** / Droniou, Jerome; Vazquez, Juan-Luis.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions

AU - Droniou, Jerome

AU - Vazquez, Juan-Luis

PY - 2009

Y1 - 2009

N2 - We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation -div(D del u) + div(V u) = f posed in a bounded domain Omega subset of R(N), with pure Neumann boundary conditions D del u . n = (V . n) u on partial derivative Omega. Under the assumption that V is an element of L(p)(Omega)(N) with p = N if N >= 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u is an element of H(1)(Omega) if f(Omega) f dx = 0, and also that the kernel is generated by a function (u) over cap is an element of H(1)(Omega), unique up to a multiplicative constant, which satisfies (u) over cap > 0 a.e. on Omega. We also prove that the equation -div(D del u) + div(Vu) +v u = f has a unique solution for all v > 0 and the map f -> u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation -div(D(T)del v) - V .del v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.

AB - We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation -div(D del u) + div(V u) = f posed in a bounded domain Omega subset of R(N), with pure Neumann boundary conditions D del u . n = (V . n) u on partial derivative Omega. Under the assumption that V is an element of L(p)(Omega)(N) with p = N if N >= 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u is an element of H(1)(Omega) if f(Omega) f dx = 0, and also that the kernel is generated by a function (u) over cap is an element of H(1)(Omega), unique up to a multiplicative constant, which satisfies (u) over cap > 0 a.e. on Omega. We also prove that the equation -div(D del u) + div(Vu) +v u = f has a unique solution for all v > 0 and the map f -> u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation -div(D(T)del v) - V .del v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.

UR - http://www.springerlink.com/content/71123gw40740j0q8/?MUD=MP

U2 - 10.1007/s00526-008-0189-y

DO - 10.1007/s00526-008-0189-y

M3 - Article

VL - 34

SP - 413

EP - 434

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 4

ER -